# What are the points of inflection, if any, of #f(x)=x+sin^2x # on [0,2pi]?

They are

The graph's points of inflection are where the concavity changes.

The concavity of this function varies at the exact location where the second derivative's sign changes.

#{: (bb "Interval", bb"Sign of "f',bb" Concavity"), ([0,pi/4)," " -" ", " "" Down"), ((pi/4, (3pi)/4), " " +, " " " Up"), (((3pi)/4 ,(5pi)/4), " " -, " "" Down"), (((5pi)/4 ,(7pi)/4), " " +, " "" Up"), (((7pi)/4 ,2pi), " " -, " "" Down") :}#

Consequently, the graph's inflection points are those that have

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To find the points of inflection of ( f(x) = x + \sin^2(x) ) on the interval ([0, 2\pi]), we need to find the second derivative of ( f(x) ) and then identify the values of ( x ) where the second derivative changes sign.

First, find the first derivative of ( f(x) ): [ f'(x) = 1 + 2\sin(x)\cos(x) ]

Now, find the second derivative of ( f(x) ): [ f''(x) = 2\cos^2(x) - 2\sin^2(x) ]

Now, determine where ( f''(x) ) changes sign. This happens when ( \cos^2(x) ) and ( \sin^2(x) ) are equal, which occurs at ( x = \frac{\pi}{4} + k\pi ) and ( x = \frac{3\pi}{4} + k\pi ), where ( k ) is an integer.

For ( x = \frac{\pi}{4} + k\pi ): [ f''\left(\frac{\pi}{4} + k\pi\right) = 2\cos^2\left(\frac{\pi}{4} + k\pi\right) - 2\sin^2\left(\frac{\pi}{4} + k\pi\right) ] [ = 2\left(\frac{1}{\sqrt{2}}\right)^2 - 2\left(\frac{1}{\sqrt{2}}\right)^2 ] [ = 0 ]

For ( x = \frac{3\pi}{4} + k\pi ): [ f''\left(\frac{3\pi}{4} + k\pi\right) = 2\cos^2\left(\frac{3\pi}{4} + k\pi\right) - 2\sin^2\left(\frac{3\pi}{4} + k\pi\right) ] [ = 2\left(-\frac{1}{\sqrt{2}}\right)^2 - 2\left(\frac{1}{\sqrt{2}}\right)^2 ] [ = 0 ]

Thus, there are no points of inflection for ( f(x) = x + \sin^2(x) ) on the interval ([0, 2\pi]).

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The points of inflection of ( f(x) = x + \sin^2(x) ) on the interval ([0, 2\pi]) are ( x = \frac{\pi}{2} ) and ( x = \frac{3\pi}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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